I had this exact same problem yesterday, when I came to pay the balance on our cruise.

Like you, I wanted to avoid a long and unnecessary phone call. Eventually, I found the option to pay on-line by logging on to my Royal Caribbean account and the payment tab was there waiting for me. I had previously set up a Royal Caribbean account from choosing 'my cruises' option in the 'manage my booking' link included in the original invoice emailed to me on booking.